section 3.3 properties of functions. testing the graph of a function for symmetry

Section 3.3

Properties of Functions

Testing the Graph of a Function for Symmetry

Symmetries

Definition: Two points (a , b) and (c , d) in the plane

are said to be symmetric about the x-axis if a c and b d symmetric about the y-axis if a c and b d symmetric about the origin if a c and b d

Symmetries

Schematically

In the above figure, P and S are symmetric about the x-axis, as are Q

and R; P and Q are symmetric about the y-axis, as are R and S; and P

and R are symmetric about the origin, as are Q and S.

Reflections

To reflect a point (x , y) about the:

x-axis, replace y with y. y-axis, replace x with x. origin, replace x with x and y with y.

Symmetries of Relations

Definition: Symmetry with respect to the x-axis.

A relation R is said to be symmetric with respect to

the x-axis if, for every point (x , y) in R, the point

(x , y) in also in R.

That is, the graph of the relation R remains the

same when we change the sign of the y-coordinates

of all the points in the graph of R.

Symmetries of Relations

Definition: Symmetry with respect to the y-axis.

A relation R is said to be symmetric with respect to

the y-axis if, for every point (x , y) in R, the point

( x , y) in also in R.

That is, the graph of the relation R remains the

same when we change the sign of the x-coordinates

of all the points in the graph of R.

Symmetries of Relations

Definition: Symmetry with respect to the origin.

A relation R is said to be symmetric with respect to

the origin if, for every point (x , y) in R, the point

( x , y) in also in R.

That is, the graph of the relation R remains the

same when we change the sign of the x- and the y-

coordinates of all the points in the graph of R.

Symmetries of Relations

In many situations the relation R is defined by an

equation, that is, R is given by

R {(x , y) | F (x , y) 0},

where F (x , y) is an algebraic expression in the

variables x and y.

In this case, testing for symmetries is a fairly

simple procedure as explained in the next slides.

Testing the Graph of an Equation for Symmetry

To test the graph of an equation for symmetry about the x-axis: substitute (x , y) into the

equation and simplify. If the result is equivalent to the original equation, the graph is symmetric about the x-axis.

about the y-axis: substitute ( x , y) into the equation and simplify. If the result is equivalent to the original equation, the graph is symmetric about the y-axis.

Testing the Graph of an Equation for Symmetry

To test the graph of an equation for symmetry about the origin: substitute ( x , y) into the

equation and simplify. If the result is equivalent to the original equation, the graph is symmetric about the origin.

Examples

Find the x- and y-intercepts (if any) of the graph of

(x 2)2 + y 2 = 1. Test for symmetry.

Examples

Find the x- and y-intercepts (if any) of the graph of

x 3 + y

33xy = 0. Test for symmetry.

Examples

Find the x- and y-intercepts (if any) of the graph of

x 4 = x

2 + y 2. Test for symmetry.

Examples

Find the x- and y-intercepts (if any) of the graph of

y 2 = x

3 + 3x 2. Test for symmetry.

Examples

Find the x- and y-intercepts (if any) of the graph of

(x 2 + y

2)2 = x 3 + y

3. Test for symmetry.

A Happy Relation

Find the x- and y-intercepts (if any) of the graph of

10/(x2 + y21) + 1/((x 0.3)2+(y 0.5)2) +

1/((x + 0.3)2 + (y 0.5)2) + 10/(100x2 + y2) +

1/(0.05x2 +10(y x2 + 0.6)2)=100

Test for symmetry.

A Happy Relation

Symmetries of Functions

When the relation G is a function defined by an

equation of the form y f (x) , then G is given by

G {(x , y) | y f (x) }.

In this case, we can only have symmetry with

respect to the y-axis and symmetry with respect to

the origin. A function cannot be symmetric with respect to the x-axis

because of the vertical line test.

Even and Odd Functions

Definition: A function f is called even if its graph

is symmetric with respect to the y-axis.

Since the graph of f is given by the set

{(x , y) | y f (x) }

symmetry with respect to the y-axis implies that

both (x , y) and ( x , y) are on the graph of f.

Therefore, f is even when f (x) f (x) for all x

in the domain of f.

Even and Odd Functions

Definition: A function f is called odd if its graph

is symmetric with respect to the origin.

Since the graph of f is given by the set

{(x , y) | y f (x) }

symmetry with respect to the origin implies that

both (x , y) and ( x , y) are on the graph of f.

Therefore, f is odd when f (x) f (x) for all x

in the domain of f.

Determine whether each graph given is an even function, an odd function, or a function that is neither even nor odd.

Even function because it is symmetric with respect to the y-axis

Neither even nor odd. No symmetry with respect to the y-axis or the origin.

Odd function because it is symmetric with respect to the origin.

Examples Using the Graph

3) 5a f x x x 35f x x x 3 5x x f x

Odd function symmetric with respect to the origin

2) 2 3b g x x 232g x x 22 3 ( )x g x

Even function symmetric with respect to the y-axis

Examples Using the Equation

Determine whether each of the following functions is even, odd function, or neither.

3) 14c h x x 34 1h x x 34 1x

Since h (– x) does not equal h (x) nor – h (x), this function h is neither even nor odd and is not symmetric with respect to the y-axis or the origin.

General Function Behavior

Using the graph to determine where the function is increasing,

decreasing, or constant

INCREASING

DECREASING

CONSTANT

Find all values of x where the function is increasing.

Example Using the Graph

The function is increasing on 4 < x < 0. That is, on (4 , 0).

Example Using the Graph

The function is decreasing on 6 < x < 4 and also on 3 < x < 6. That is, on (6 ,

4 ) (3 , 6 )

Find all values of x where the function is decreasing.

Example Using the Graph

The function is constant on 0 < x < 3. That is, on (0 , 3).

Find all values of x where the function is constant.

Precise Definitions

Suppose f is a function defined on an interval I. We

say f is: increasing on I if and only if f (x1) < f (x2) for all

real numbers x1, x2 in I with x1 < x2.

decreasing on I if and only if f (x1) > f (x2) for all real numbers x1, x2 in I with x1 < x2.

constant on I if and only if f (x1) f (x2) for all real numbers x1, x2 in I.

General Function Behavior

Using the graph to find the location of local maxima and local minima

Local Maxima

The local maximum is f (c) and occurs at x c

Local Minima

The local minimum is f (c) and occurs at x c

Precise Definitions

Suppose f is a function with f (c) d.

We say f has a local maximum at the point (c , d) if

and only if there is an open interval I containing c

for which f (c) f (x) for all x in I.

The value f (c) is called a local maximum value of

f and we say that it occurs at x c.

Precise Definitions

Suppose f is a function with f (c) d.

We say f has a local minimum at the point (c , d) if

and only if there is an open interval I containing c

for which f (c) f (x) for all x in I.

The value f (c) is called a local minimum value of

f and we say that it occurs at x c.

There is a local maximum when x = 1.

The local maximum value is 2

Example

There is a local minimum when x = 1 and x = 3.

The local minima values are 1 and 0.

Example

(e) The region where f is increasing is.

(f) The region where f is decreasing is.

1,1 3,

, 1 1,3

Example

General Function Behavior

Using the graph to locate the absolute maximum and the

absolute minimum of a function

Absolute Extrema

Absolute Minimum

Let f be a function defined on a domain D and let c be in D.

Absolute Maximum

The number f (c) is called the absolute maximum value of f in D and we say that it occurs at c.

A function f has an absolute (global) maximum at x c in D if f (x) f (c) for all x in the domain D of f.

Absolute Maximum

Absolute Extrema

c

( )f c

Absolute Minimum

Absolute Extrema

A function f has an absolute (global) minimum at x c in D if f (c) f (x) for all x in the domain D of f.The number f (c) is called the absolute minimum value of f in D and we say that it occurs at c.

c

( )f c

Find the absolute extrema of f, if they exist.

The absolute maximum of 6 f (3) occurs when x 3.

The absolute minimum of 1 f (0) occurs when x 0.

Examples of Absolute Extrema

The absolute maximum of 3 f (5) occurs when x = 5.

There is no absolute minimum because of the “hole” at x = 3.

Examples of Absolute Extrema

Find the absolute extrema of f, if they exist.

The absolute maximum of 4 f (5) occurs when x = 5.

The absolute minimum of 1 occurs at any point on the interval [1,2].

Examples of Absolute Extrema

Find the absolute extrema of f, if they exist.

There is no absolute maximum.

The absolute minimum of 0 f (0) occurs when x = 0.

Examples of Absolute Extrema

Find the absolute extrema of f, if they exist.

There is no absolute maximum.

There is no absolute minimum.

Examples of Absolute Extrema

Find the absolute extrema of f, if they exist.

Extreme Value Theorem

If a function f is continuous* on a closed interval [a, b], then f has both, an absolute maximum and absolute minimum on [a, b]. Moreover each absolute extremum occurs at a local extrema or at an endpoint.

*Although it requires calculus for a precise definition, we will agree for now that a continuous function is one whose graph has no gaps or holes and can be traced without lifting the pencil from the paper.

a b a ba b

Attains both a max. and min.

Attains a min. but no a max.

No min. and no max.

Open Interval Not continuous

Extreme Value Theorem

Using a Graphing Utility to Approximate Local Extrema

3Use a graphing utility to graph 2 3 1 for 2 2.

Approximate where has any local maxima or local minima.

f x x x x

f

Example

Example

3Use a graphing utility to graph 2 3 1 for 2 2.

Determine where is increasing and where it is decreasing.

f x x x x

f

Average Rate of Changeof a Function

Delta Notation

2 1q q q

If a quantity q changes from q1 to q2 , the change in q is denoted by q and it is computed as

Example: If x is changed from 2 to 5, we write

2 1 5 2 3x x x

Delta Notation

2 1q q q

If a quantity q changes from q1 to q2 , the change in q is denoted by q and it is computed as

Note: In the definition of q we do not assume that q1 < q2. That is, q may be positive, negative, or zero depending on the values of q1 and q2.

Delta Notation

Example - Slope: the slope of a non-vertical line that passes through the points (x1 , y1) and (x2 , y2) is given by:

2 1

2 1

y yym

x x x

change in

change in

f f

x x

( ) ( )f b f a

b a

The average rate of change of f (x) over the interval [a , b] is

The change of f (x) over the interval [a , b] is

Difference Quotient

( ) ( )f f b f a

Average Rate of Change

a) From 1 to 3, that is, over the interval [1 , 3].

Average Rate of Change

b) From 1 to 5, that is, over the interval [1 , 5].

Average Rate of Change

c) From 1 to 7, , that is, over the interval [1 , 7].

Average Rate of Change

Geometric Interpretation of the Average Rate of Change

, ( )P a f a

, ( )Q b f b

secant line has slope mS

( ) ( )S PQ

f f b f am m

x b a

Is equal to the slope of the secant line through the points (a , f (a)) and (b , f (b)) on the graph of f (x)

x b a

( ) ( )f f b f a

Average Rate of Change

Example: Compute the average rate of change of over [0 , 2].2( ) 3 2f x x x

( ) ( )

(2) (0)

2 0

S PQf f b f a

m mx b a

f f

8 0

2 0

= 4

This means that the slope of the secant line to the graph of parabola through the points (0 , 0) and (2 , 8) is mS 4.

Average Rate of Change

Geometric Interpretation

8f

2x

Q

P

The units of f , change in f , are the units of f (x).

The units of the average rate of change of f are units of f (x) per units of x.

units of ( )is given in

units of

f f x

x x

Units for f x

Example

The graph on the next slide shows the numbers of

SUVs sold in the US each year from 1990 to 2003.

Notice t 0 represents the year 1990, and N(t)

represents sales in year t in thousands of vehicles.

a) Use the graph to estimate the average rate of change of N(t) with respect to t over [6 , 11] and interpret the result.

b) Over which one-year period (s) was the average rate of N(t) the greatest?

Example

a) The average rate of change of N over [6 , 11] is given by the slope of the line through the points P and Q.

Example

P

Q

N

t

P

Q

N

t

thousands SUVsyear

(11) (6) 20011 6

N N Nt

Compute

Sales of SUVs were increasing at an average rate of 200,000 SUVs per year from 1996 to 2001.

P

Q

N

t

Interpret

Suppose that g (x) 2x2 + 4x 3.

a) Find the average rate of change of g from 2 to 1.

b) Find the equation of the secant line containing the points on the graph corresponding to x 2 and x 1.

c) Use a graphing utility to draw the graph of g and the secant line on the same screen.

Equation of the Secant line

Suppose that g (x) 2x2 + 4x 3.a) Find the average rate of change of g from 2 to 1.

Solution

1 19(1) ( 2)

1 2 3

18 6

3

y g g

x

b) Find the equation of the secant line containing the points on the graph corresponding to x 2 and x 1.

From part a) we have m 6. Using the point-slope

formula for the line we have

Equation of the Secant line

1 1

( 1

9)

6( ( 2))

19 6 12

( )y y m x

x

y x

x

y

6or 7y x

c) Use a graphing utility to draw the graph of g and the secant line on the same screen.

Equation of the Secant line